Sum of Larger Ideals is Larger

Theorem

Let $R$ be a commutative ring with unity.

Let $\mathfrak a_1, \mathfrak a_2, \mathfrak b_1, \mathfrak b_2$ be ideals of $R$ such that:

$\mathfrak a_1 \subseteq \mathfrak a_2$

and:

$\mathfrak b_1 \subseteq \mathfrak b_2$

Then their sums satisfy:

$\mathfrak a_1 + \mathfrak b_1 \subseteq \mathfrak a_2 + \mathfrak b_2$

Proof

Let $x \in \mathfrak a_1 + \mathfrak b_1$.

By Definition of Sum of Ideals, there are $a \in \mathfrak a_1$ and $a \in \mathfrak b_1$ such that:

$x = a + b$

Then:

$a \in \mathfrak a_1 \subseteq \mathfrak a_2$

and:

$b \in \mathfrak b_1 \subseteq \mathfrak b_2$

Thus, by Definition of Sum of Ideals:

$x \in \mathfrak a_2 + \mathfrak b_2$

$\blacksquare$